Journal of Graph Algorithms and Applications
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Intersection Graphs of Pseudosegments: Chordal Graphs
Cornelia Dangelmayr, Stefan Felsner, and William T. Trotter
Vol. 14, no. 2, pp. 199-220, 2010. Regular paper.
Abstract We investigate which chordal graphs have a representation as intersection graphs of pseudosegments. For positive we have a construction which shows that all chordal graphs that can be represented as intersection graphs of subpaths on a tree are pseudosegment intersection graphs. We then study the limits of representability. We identify certain intersection graphs of substars of a star which are not representable as intersection graphs of pseudosegments. The degree of the substars in these examples, however, has to be large. A more intricate analysis involving a Ramsey argument shows that even in the class of intersection graphs of substars of degree three of a star there are graphs that are not representable as intersection graphs of pseudosegments. Motivated by representability questions for chordal graphs we consider how many combinatorially different k-segments, i.e., curves crossing k distinct lines, an arrangement of n pseudolines can host. We show that for fixed k this number is in O(n2).
Submitted: October 2008.
Reviewed: December 2009.
Revised: December 2009.
Accepted: December 2009.
Final: December 2009.
Published: February 2010.
Communicated by Henk Meijer